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A double Legendre polynomial quadrature-free method was presented to analyze axisymmetric free vibration problem for axially polarized Functionally Graded Piezoelectric circular plates. The present analytical method directly incorporated the boundary conditions into the vibration control equation. At the same time, the mechanical displacements and the electrical potential were defined as the form of double Legendre polynomials series. Then, the solution of eigenvalue was used to obtain the resonant/anti-resonant frequencies. In view of including lots of time-consuming integration calculations, the quadrature-free process was deduced to improve the computational efficiency. The valid of the presented method was verified by the comparison with the existing literature and the commercial software COMSOL. Furthermore, the effects of the material gradient index, radius-thickness ratio on the frequencies, and electro-mechanical coupling efficient were studied. The results showed that the frequency-radius product tends to be stable with the increase of the radius-thickness ratio, and the increase of gradient index led to the increase of the frequency-radius product. [ABSTRACT FROM AUTHOR]

The paper presents a semi-analytical method, named the Double Legendre Polynomial Method (DLPM), which is extended for modeling the axisymmetric free vibration of a functionally graded piezoelectric (FGP) hollow cylinder resonator. The FGP resonator is composed of two-phase graded piezoelectric materials, Lead zirconate titanate (PZT4) and aluminum nitrid (AlN), on the top and bottom surfaces, respectively, where the material properties gradually change along the thickness direction. The integration of FGP materials with piezoelectric materials is an innovative aspect in material science, and this paper explores their dynamic behavior, providing insights into their mechanical and electrical properties and contributes to expand our understanding of the axisymmetric free vibration of FGP hollow cylinder resonators as a crucial factor for optimizing their performance across a wide range of applications, including sensors, actuators, energy harvesting, and structural health monitoring systems. The DLPM is applied to solve the wave governing differential equations of motion, offering a new tool for studying the dynamic behavior of FGP hollow cylinders under axisymmetric vibration. The study investigates the effects of the diameter-thickness ratio on the effective electromechanical coupling coefficient and the resonant and anti-resonant frequencies of the FGP hollow cylinder. The results show that as the diameter-thickness ratio increases, the frequency-diameter product appears to become more stable. This observation is a valuable insight for optimizing the design of these resonators. Furthermore, the study presents the electrical input admittance, the natural frequencies as well as the mechanical displacement and electric potential profiles of the resonator. All the provided results are based on the calculation of the resonant frequencies. The validity of the presented results was verified by the comparison with the existing ones reported in the literature. [ABSTRACT FROM AUTHOR]

The materials industry is under increasing pressure to enhance performance while lowering costs and environmental impact. Three layers of functionally graded materials (FGMs) were proposed as a solution to this problem. The general goal of this research is to develop a semi-analytical method for solving the free vibration problem of FGMs in an infinite-length composite cylinder made of stainless steel (SS) and silicon nitride (SN). The formulation is based on Legendre polynomial approach and the use of specific rectangle window functions. This approach will result in an eigenvalue/eigenvector issue in the computation of normalized frequencies and displacement profiles. The effective material properties of FGM cylinders vary following the radial direction in conformity with the power-law distribution of the volume fractions. The numerical iteration approach was generated to create data utilizing MATLAB apps to investigate the free vibration analysis from specified boundary conditions. The results are obtained for longitudinal, torsional, and flexural modes. Then, the propagation of guided waves in three-layers (SN/SS/SN) of functionally graded cylinder with diverse gradient for each mode is studied. Accordingly, the results obtained demonstrate that the volumetric fraction of the FGM cylinder is significantly impacted by the graded-index. In addition, the graded-index has a major influence on the variation of the material's properties in the radial direction. Furthermore, the dispersion curves of the normalized frequencies and phase velocities are significantly influenced by the graded-index. These results demonstrate that the phase velocities of the same mode increase as the exponents of the power-law increase. The displacement distributions are also explored to better understand the characteristics of the guiding waves inside the materials. Furthermore, the boundary conditions have a more significant influence on the normal stresses. High accuracy is found between our results and those published in the literature (error = 0.58%). This paper provides an important insight into the propagation of ultrasonic-guided waves in functionally graded multilayer cylindrical constructs for non-destructive testing. [ABSTRACT FROM AUTHOR]

This article presents an extension of the double Legendre polynomial method to model and evaluate the axisymmetric free vibration problem for the axially polarized piezoelectric resonator. The aim is to show the benefit of the employed method. The polynomial series was used to define the mechanical displacements and electrical potential at the same time. The proposed approach considerably improves computing efficiency by avoiding much of the numerical integration calculations. As a result, the electromechanical coupling coefficient shows that the fundamental mode is the most piezoactive mode in the structure. Furthermore, the electrical input impedance and field profiles are easily presented. [ABSTRACT FROM AUTHOR]

Due to the limit of preparation technology, defects, such as cracks and inclusions, are inevitable in the functionally graded quasi‐crystal material. For the purpose of the guided wave non‐destructive testing on the functionally graded (FG) two‐dimensional(2‐D) hexagonal quasi‐crystal(QC) plates, Lamb and SH wave characteristics in the context of the Bak's model are investigated by means of the Legendre polynomial method. The convergence of the polynomial method applied on the FG 2‐D hexagonal QC plate is analyzed. Some new wave characteristics resulted from the phonon‐phason coupling effect are revealed. There are three Lamb‐like modes without cut‐off frequencies for the x‐y and y‐z QC plates, but four Lamb‐like modes without cut‐off frequencies for the x‐z QC plate. For phonon modes, phonon displacements and stresses are bigger than phason displacements and stresses. However, they are just opposite for phason modes. It can be utilized to distinguish phonon modes and phason modes. [ABSTRACT FROM AUTHOR]

Functionally graded piezoelectric–piezomagnetic (FGPP) material simultaneously consists of piezomagnetic and piezoelectric phases, which are able to convert energy among mechanical, electric, and magnetic fields. The magneto-electric effect on waves in FGPP fan-shaped cylindrical structures is studied by exploiting the double Legendre orthogonal polynomial method. By means of the Heaviside function, the initial conditions are brought into wave motion equations. Dispersion properties, electric and magnetic potential, and the Poynting vector are calculated. Subsequently, the effect of the graded variation and geometric size on wave characteristics is analyzed. The FGPP fan-shaped cylindrical structures are of complex geometrical shape and material inhomogeneity, so their influences on the magneto-electric effect are the focus of discussion. Results reveal that the cut-off frequencies have a negative relationship with the cross-section area of the structure. The magneto-electric effect could be adjusted via altering the geometric size of the cross-section. These results can be utilized to design and optimize piezoelectric–piezomagnetic fan-shaped transducers. [ABSTRACT FROM AUTHOR]

The Legendre polynomial approach has been proposed to solve wave propagation in multilayered flat plates and functionally graded structures for more than ten years, but it can deal with a multilayered plate only when the material properties of two adjacent layers do not change significantly. In this paper, an improvement of the Legendre polynomial approach is proposed to solve wave propagation in what, from now on, we will call general multilayered piezoelectric cylindrical plates, to mean indifferently with or without very dissimilar materials. Detailed formulations are given to highlight the differences from the conventional Legendre polynomial approach. Through numerical comparisons among the exact solution (from the reverberation-ray matrix), the conventional polynomial approach, and the improved polynomial approach, the validity of the proposed approach is illustrated. Then, the influences of the radius-to-thickness ratio on the dispersion curves, the stress, and electric displacement distributions are discussed. It is shown that the conventional orthogonal polynomial approach cannot obtain correct continuous normal stress and normal electric displacement shapes, unlike the improved orthogonal polynomial approach, which overcomes these drawbacks. It is also found that three factors determine the distribution of mechanical energy and electric energy at higher frequencies: the radius-to-thickness ratio, the wave speed of component material, and the position of the component material. [ABSTRACT FROM PUBLISHER]